Properties of generalized Schwarzschild spacetimes with extra dimensions

Abstract

We show that an ansatz for $1+3+n$ dimensional static spacetime with spherical symmetry in three dimensions and Euclidean symmetry in $n$ dimensions, parametrized by only one function of radial coordinate, leads to a limited set of vacuum solutions of the Einstein field equations. They can also be identified as Weyl solutions. We investigate properties of these spacetimes through the Kretschmann scalar, Newtonian mass defined through the Newtonian limit, Komar mass, Einstein, Landau--Lifshitz, and ADM mass. In addition to $1+3+n$ dimensional Minkowski spacetime, there are two classes of solutions. The first class is a trivial product of the Schwarzschild spacetime and Euclidean spaces in extra dimensions, while the second class is non-trivial. In the case with no horizon, there is a naked singularity, all masses are equal, and they are negative. In the case when there is a horizon, this horizon accommodates a physical singularity, which corresponds to Kaluza--Klein bubbles featuring exotic properties. Einstein, Landau--Lifshitz, and ADM masses are positive, while Newtonian and Komar masses are negative. This differentiates these solutions from trivial higher-dimensional extensions of the Schwarzschild solution.